The present invention relates to a method of forming a semiconductor structure.
Pair-production by impact ionisation is one of the most important processes affecting the performance of semiconductor electronic devices. The secondary carriers lead to current multiplication which is used to increase the signal in avalanche photodiodes and phototransistors. However, avalanche breakdown imposes an upper limit on the bias that can be applied to semiconductor pn junctions, for example in diodes and transistors, which limits the power available from such devices.
In many cases it is desirable to control the likelihood of impact ionisation and avalanche breakdown by means of a suitable choice of semiconductor material, and/or by engineering the electric field profile by means of suitable doping with p- or n-type impurities. Additionally, the conduction band and valence band-structure may be tailored by means of compositional variation (i.e. alloying) or by growth of heterojunctions comprising layered materials of different band-structure. Lattice mismatched systems may be grown and the resulting strain used to modify the material properties.
The design of semiconductor structures with controlled ionisation or avalanche breakdown properties requires an understanding of the effects of the material properties on the ionisation coefficients. In particular, the energy band-structure is known to have a significant effect on impact ionisation (F. Capasso, Physics of Avalanche Photodiodes, Academic, San Diego, 1985). Many attempts have been made to design and engineer structures with artificially enhanced or suppressed ionisation coefficients, but these have been rather unsuccessful and considerable controversy remains concerning the validity of the experimental results.
The failure of these previous attempts has been due to an incorrect understanding of the effect of the band-structure on the ionisation rates and, hence, a use of incorrect or inaccurate prescriptions for the design of devices. As an example, we refer to the prescriptions given in Sze S. M. Physics of Semiconductor Devices 2nd Edition (Wiley, 1981) at page 104 for the breakdown voltage Vb in abrupt p-n junctions and linearly graded junctions.
These are reproduced below.
Abrupt junction:
Vb=60(Eg/1.1)3/2 (NB/1016)xe2x88x923/4xe2x80x83xe2x80x83(1)
Linearly graded junction:
Vb=60(Eg/1.1)6/5 (a/3xc3x971020)xe2x88x922/5xe2x80x83xe2x80x83(2)
The parameters Eg, NB and a are the bandgap the background doping density and the doping gradient respectively. These prescriptions were based on the experimentally measured ionisation coefficients of Ge, Si, GaAs and GaP. A similar prescription has been calculated which applies to p-i-n diodes with a 1 xcexcm thick i-layer:
Vb=30(Eg/1.1)xe2x80x83xe2x80x83(3)
Although, the apparent linear dependence of Vb on the band-gap Eg holds approximately for these materials, it has not been known whether the relation holds for other materials, or how accurate the expression is. Nevertheless, these prescriptions have been widely used and applied to many materials. Furthermore, the basic assumption contained in these formulae, viz. that it is the energy bandgap Eg that primarily determines the ionisation coefficients, is the assumption common to apparently all the previous attempts to control and engineer ionisation coefficients.
Other simple theories of impact ionisation have expressed ionisation coefficients in terms of ionisation threshold energies, phonon energies, and the electron-phonon scattering mean-free path, with the latter two variables treated as parameters which are varied to fit experimental data for the ionisation coefficients [for a detailed review, see F. Capasso, in Semiconductors and Semimetals (ed. W. T. Tsang, Academic, New York) 22D (1986) 1]. Hence these theories have no predictive power in respect of the ionisation characteristics of different materials.
Recently, considerable progress has been made in the first-principles numerical calculation of ionisation coefficients, incorporating the effects of the full bandstructure on the carrier kinematics [Shichijo and Hess, Phys. Rev. B 23, p.4197, 1981], scattering dynamics [M. V. Fischetti and S. E. Laux, Phys. Rev. B38, 9721 (1988)] and impact ionisation cross-section [N. Sano and A. Yoshii, Phys. Rev. B45 (1992) 4171; J. Bude and K. Hess, J. Appl. Phys. 72 (1992) 3554, Y. Kamakura, H. Mizuno, M. Yamaji, M. Morifuji, K. Taniguchi, C. Hamaguchi, T. Kunikiyo, M. Takenaka, J. Appl. Phys. 75 (1994) 3500]. However, few materials have been studied to date. Due to the numerical complexity, no simple relation between the ionisation coefficients or breakdown voltage and energy bandstructure has been expected. Selection of semiconductor materials with desired breakdown properties has therefore been made based on empirical knowledge of the ionisation rates in each material.
Optimisation of semiconductor device performance may impose apparently contradictory requirements on the properties of the constituent materials. For example, high speed operation of a field-effect transistor (FET) requires a material with high transient electron velocity (implying a low electron effective mass and hence narrow bandgap), whereas high power operation requires a high breakdown voltage (previously thought to imply a material with wide bandgap). Similarly, a photodetector for use in the near infra-red wavelength region conventionally requires a material with a narrow direct bandgap, whereas the requirements of low dark current suggest the use of a material with a wide bandgap to reduce both primary generation of dark current and its subsequent multiplication by impact ionisation. Some of these trade-offs have in the past been addressed by using composite structures, in which layers of different material composition perform different functions.
The effect of impact ionisation and breakdown, and hence the ionisation properties desired, are different for different classes of device. Here we summarise design optimisation problems in respect of these properties in Field-Effect Transistors (FETs), bipolar transistors, avalanche photodiodes (APDs) and Impact Ionisation Avalanche Transit Time (IMPATT) diodes.
FETs:
High-speed FET operation requires a material in which carriers have a high effective velocity, and whose transport can be efficiently modulated by the gate. This generally implies a material with a narrow energy bandgap. However, high-power operation requires a high avalanche breakdown voltage which previously has been identified with a wide bandgap.
Known ultra-short gate length FETs have cut-off frequencies well above 100 GHz. Many microwave and millimetre-wave applications in optical communications, for example laser drivers and photoreceiver amplifiers, and radio, for example oscillators and low-noise amplifiers, require high-speed devices capable of handling large currents and voltages.
The highest operation frequencies have been obtained in AlInAs/GaInAs high electron mobility transistors (HEMTs). However, the narrow bandgap of InGaAs and the consequent high probability of impact ionisation and Zener tunnelling leads to several problems, including low source-drain breakdown voltage and the presence of xe2x80x9ckinkxe2x80x9d phenomena.
The output power of FETs with GaAs and InGaAs channels is limited by avalanche breakdown. In order to increase the power, heterojunction FETs with doped InP channels have been fabricated. [D. R. Greenberg, J. A. del Alamo and R. Bhat IEEE Trans. Electron Devices, ED 42 1574(1995)]. The smaller ionisation probability of InP, compared with GaAs or InGaAs, leads to increased breakdown voltage and hence voltage swing. A complete absence of ionisation in the channel has been achieved hitherto, with consequently low gate current and low output conductance. However, the cut-off frequency was reduced due to the lower mobility of InP and to the doping in the channel.
To achieve high-frequency and high-power operation of FETs, devices with composite InGaAs/InP channels have been proposed and demonstrated [T. Enoki, K. Arai, A. Kohzen and Y. Ishii, IEEE Trans. Electron Dev. 42, 1413 (1995)]. Electrons are confined to the high-mobility InGaAs in the low-field regions of the channel, and hence exhibit a high transient velocity. However, when the field is sufficiently high, the carriers undergo real-space transfer into the adjoining InP sub-channel, which exhibits high saturated velocity and high breakdown voltage. Hence the output power is not limited by breakdown in the InGaAs. In this configuration, the scope for independent optimisation of the low-field transport and high-field breakdown is somewhat limited by the requirement that real-space transfer into the InP layer should occur prior to the onset of impact ionisation in the InGaAs channel.
Bipolar Transistors:
The output power of bipolar transistors is limited by the breakdown voltage of the base-collector junction, and the operating frequency by the base-collector transit time. As bipolar transistors are reduced in size, high doping is required in the base and collector to reduce parasitic resistances and base punch-through. The increased field in the base-collector junction leads to impact ionisation and hence base current reversal and xe2x80x9csnap-backxe2x80x9d phenomena [L. Vendrame, E. Zabotto, A. Dalfabbro, A. Zanini, G. Verzellesi, E. Zanoni, A. Chantre and P. Pavan, IEEE Trans. Electron Dev. 42, 1639 (1995)]. Further, heterojunction bipolar transistors HBTs) operating in avalanche mode exhibit higher collector-base junction capacitance, lower Early voltage, higher device noise, lower power efficiency, lower cut-off frequency, reduced device switching speed, and degraded maximum frequency of oscillation [J. S. Yuan, Int. J. Electronics 74, 909 (1993)]. Hence it is necessary to find materials for the collector in which the ionisation probability is reduced, while the saturation velocity remains high or is even increased.
HBTs with wide-bandgap emitters offer improved characteristics such as higher emitter efficiency due to reduced hole injection from the base, decreased base resistance, etc., which lead to improved gain and bandwidth [see S. M. Sze, Physics of Semiconductor devices, (Wiley, N.Y., 1981), Ch. 3]. In conventional AlGaAs/GaAs, AlInAs/GaInAs or InP/GaInAs devices, impact ionisation in the GaAs or GaInAs collector limits the output power. The situation is improved using double heterostructure devices in which the collector is fabricated from a material with lower ionisation coefficients. AlInAs/GaInAs/InP devices with InP collectors exhibited breakdown voltage≈3 times higher than devices with an InGaAs collector, although with ft reduced by about 30% [C. W. Farley, J. A. Higgens, W. J. Ho, B. T. McDermott and M. F. Chang, J. Vac. Sci, Technol. B 10, 1023 (1992)]. The energy barrier at the InGaAs/InP interface requires the use of intermediate grading layers to maintain the high-speed operation [H. F. Chau, D. Pavlidis, J. Hu and K. Tomizawa, IEEE Trans. Electron Dev. 40, 2 (1993)]. More sophisticated designs incorporate an InGaAs/AlInAs chirped superlattice and a doping dipole at the base-collector junction, allowing efficient injection into the InP collector, and exhibit high output power at microwave frequencies [C. Nguyen, T. Y. Liu, M. Chen, H. C. Sun and D. Rensch, IEEE Electron Dev. Lett. 17, 133 (1996)].
GaAs-based HBTs with InGaP collectors exhibit higher breakdown voltage compared to GaAs or AlGaAs collectors [C. R. Abernathy, F. Ren, P. W. Wisk, S. J. Pearton and R. Esagui, Appl. Phys. Lett. 61, 1092 (1992)]. Thin highly-doped GaAs or GaInP layers in the collector were required to maintain the current saturation characteristics [J. I. Song, C. Caneau, W. P. Hong and K. B. Chough, Electron. Lett. 29, 1881 (1993)].
APDs:
Impact ionisation is the direct cause of avalanche multiplication and hence photocurrent gain in APDs. Hence, sufficiently high ionisation coefficients are required so that avalanche breakdown occurs at sufficiently low fields (e.g. before breakdown due to Zener tunnelling). The penalty for large photocurrent multiplication is excess noise due to the stochastic nature of the ionisation process. The carrier avalanche leads to a finite build-up time of the photocurrent response, and to a sensitive dependence on the applied bias. The noise factor, build-up time and gain stability are all optimised when there is a large difference between the ionisation coefficients of electrons (xcex1) and holes (xcex2). For a full discussion, see Capasso [ibid]. Low-noise operation also requires a low dark current (e.g. due to band-to-band tunnelling).
The selection of materials for APDs is constrained by the desired optical wavelength for detection, which is a function of bandgap. Unfortunately, the common direct band-gap III-V materials such as GaAs, InP and InGaAs have equal (within a factor≈2) electron and hole ionisation rates, and hence high excess noise. Narrow bandgap materials such as InGaAs, which are sensitive in the technologically-important near-infrared region, suffer from large leakage currents at high fields due to band-to-band tunnelling. Only silicon exhibits unambiguously a large difference between electron and hole ionisation rates, but its absorption coefficient at visible/infrared wavelengths is small due to its indirect bandgap.
SAM APDs
In the Separate Absorption and Multiplication (SAM) APD, one layer such as InGaAs, absorbs incoming light, while carrier multiplication occurs in an adjacent high-field layer composed of a material such as InP [O. K. Kim, S. R. Forrest, W. A. Bonner and R. G Smith, Appl. Phys. Lett. 39, 402 (1981)].
Careful control of the doping is required so that the field at the heterointerface between the layers is sufficient to ensure collection of the carriers but not high enough to allow tunnelling in the InGaAs. The abrupt heterointerface causes pile-up of photogenerated holes; this deleterious effect can be ameliorated by compositional grading of the heterointerface. SAM APD""s with a Si multiplication region and SiGe superlattice region absorbing light at 1.3 xcexcm have also been demonstrated. The low absorption coefficient necessitated a waveguide geometry to increase the absorption length without drastically increasing the transit time [H. Temkin, A. Anredsyan, N. A. Olsson, T. P. Pearsall and J. C. Bean, Appl. Phys. Lett. 49, 809 (1996)].
Bandgap Engineering for Enhanced Ionisation Rates in APDs
Since no material has been found which exhibits large multiplication, low dark current, high absorption coefficient over a range of optical wavelengths, is and significantly differing electron and hole ionisation coefficients, many attempts have been made to design and engineer APDs with modified (usually enhanced) ionisation rates. The assumption common to these attempts is that the energy bandgap Eg primarily determines the ionisation coefficients. Control is attempted by varying either the bandgap itself, or the bandstructure (e.g. effective mass, spin-orbit splitting, etc.) in a restricted region of the Brillouin zone such as the region immediately in the vicinity of the local conduction-band minimum defining the band-gap, or the heterojunction discontinuities appropriate to a restricted region of the Brillouin zone such as the region in the vicinity of the bandgap. Previously-investigated structures include the graded-gap APD, multiple quantum well or superlattice APD and the staircase photomultiplier [for a review, see F. Capasso, ibid].
Graded-gap APD:
Kroemer [H. Kroemer, RCA Rev. 18, 332 (1957)] showed that electrons and holes in a compositionally-graded semiconductor experience quasi-electric fields whose force pushes the carriers in the same direction towards regions of lower bandgap. Capasso [F. Capasso, U.S. Pat. No. 4,383,269 (1983)] described a p+-i-n+ APD in which the i-region consists of a graded-gap semiconductor. An applied electric field accelerates the electrons and holes in opposite directions. Hence electrons and holes experience different electric fields, and therefore different ionisation coefficients. Experiments on graded-gap AlGaAs structures suggested some enhancement of (xcex1/xcex2) compared to GaAs [F. Capasso, W. T. Tsang, A. L. Hutchinson, and P. W. Foy, Inst. Phys. Conf. Ser 63, 473 (1992)]. Ionisation occurs at lower fields in the narrow-gap regions of the graded-gap APD than in the wide-gap regions, leading to more gradual breakdown and hence more stable gain.
Multiple Quantum Well APD:
Chin et al. [R. Chin, N. Holonyak and G. E. Stillman, Electron. Lett. 16, 467 (1980)] predicted that ionisation rates could be enhanced in semiconductor multiple quantum wells (MQWs) or superlattices. Carriers crossing a heterojunction from a wide-gap to a narrow-gap material increase their kinetic energy by an amount equal to the potential discontinuity. The high energy tail of the carrier distribution, and hence the number of carriers at energies where ionisation occurs, is increased. At a heterojunction in which the potential discontinuities for electrons and holes are significantly different, one carrier type can be selectively enhanced leading to an increase in the ionisation coefficient ratio. Hence the noise, response time and gain stability can be improved. The spatial localisation of the ionisation event within a short distance of the heterojunctions also further decreases the multiplication noise.
Capasso et al. [F. Capasso, W. T. Tsang, A. L. Hutchinson and G. F. Williams, Appl. Phys. Lett. 40, 38 (1982)] reported (xcex1/xcex2)≈8 in 500 xc3x85 AlGaAs/500 xc3x85 GaAs MQW APD FIG. 4c), compared to (xcex1/xcex2)≈2 in bulk GaAs. It should be noted that these measurements, and the measurements of enhanced (xcex1/xcex2) in graded-gap APDs mentioned above, were performed before the definitive experimental study of ionisation coefficients in GaAs by Bulman et al.[G. E. Bulman, V. M. Robbins, K. F. Brennan, K. Hess and G. E. Stillman, IEEE Electron. Dev. Lett. EDL-4, 181 (1983)], which pointed out the role of electroabsorption of recombination radiation in contaminating measurements of ionisation coefficients. More recent measurements [P. Aristin, A. Torabi, A. K. Garrison, H. M. Harris and C. J. Summers, Inst. Phys. Conf. Ser. 120, 523 (1991); A. Salokatve, M. Toivonen and M. Hovinen, Electron. Lett. 28, 416 (1992)] have found no difference between ionisation coefficients in the GaAs wells of the MQW, and bulk GaAs. In contrast, InAlAs/InGaAs superlattice APDs have exhibited clear evidence of (xcex1/xcex2) enhancement, and exhibit gain-bandwidth products  greater than 100 GHz [K. Taguchi, K. Makita, I. Watanabe, M. Tsuji and S. Suguo, Optoelectronicsxe2x80x94Devices and Technol. 10, 97 (1995)].
Staircase Photomultiplier:
The performance of the MQW APD is expected to be optimised for electron multiplication when the valence-band discontinuity is minimised, and the conduction-band is larger than the bandgap (xcex94Ec greater than Eg). However in such a case, compositional grading of the well-barrier exit heterojunction is required to prevent electron trapping in the quantum wells. These considerations led to the proposal of the staircase APD [G. F. Williams, F. Capasso and W. T. Tsang, IEEE Electron. Dev. Lett. EDL-3, 71 (1982)]. Due to the single carrier-type multiplication and the spatial localisation of the impact ionisation, the device acts as a solid state analogue of the photomultiplier. Material systems which have been identified as satisfying (or almost satisfying) the condition xcex94Ec greater than Eg include AlAs0.03Sb0.97/GaSb and HgTe/CdTe [F. Capasso, in Semiconductors and Semimetals (ed. W. T. Tsang, Academic, New York) 22D (1986) 1].
Staircase APD""s which incorporate multiple graded layers but do not satisfy the condition xcex94Ec greater than Eg, have been fabricated using GaAs/Al0.45Ga0.55As with doping dipoles at the heterojunction interfaces to increase the effective band offset [M. Toivonen, A. Salokatve, M. Hovinen, AND M. Pessa, Electron. Lett. 28, 32 (1992)], and also using InAlGaAs with xcex94Ec up to 0.5 eV [M. Tsuji, K. Makita, I. Watanabe and K. Taguchi, Jap. J. Appl. Phys. 34, L1048 (1995)].
In both cases, (xcex1/xcex2) was increased from ≈2 in the bulk materials to ≈5. Staircase APDS fabricated from amorphous hydrogenated SiC/Si [K. Sawada, S. Akata, T. Takeuchi and T. Ando, Appl. Phys. Lett. 68, 1835 (1996)], and amorphous hydrogenated SiC/SiGe multilayer APDs in which the conduction-band discontinuity exceeds the bandgap [S. Sugawa, A. Kozuka, T. Atoji, H. Tokunaga, H. Shinizu and K. Ohmi, Jap. J. Appl. Phys. 35, 1014 (1996)], have been reported.
IMPATTs:
IMPATT diodes depend on the time delay caused by avalanche build-up of secondary carriers and their subsequent drift, which can yield a negative resistance in the diode characteristic at microwave frequencies. Such devices offer the highest power sources of microwave and millimeter-wave radiation [for detailed discussion and references, see S. M. Sze, Physics of Semiconductor devices, (Wiley, N.Y., 1981), Ch. 10]. The power-frequency limit of the IMPATT diode depends on the saturated carrier velocity vs and the avalanche breakdown voltage Vb. The frequency is also limited by the avalanche build-up time and hence depends on the ratio of electron to hole ionisation coefficients. Output efficiency is maximised when a high carrier velocity can be maintained at low fields. In contrast to the case for APDs, the minimum noise in an IMPATT diode is achieved when electron and hole coefficients are similar in magnitude, so that GaAs (xcex1/xcex2≈2) offers lower noise than Si (xcex1/xcex2≈10) [Sze, ibid].
Various junction structures are utilised, including Read, double-rift and Misawa diodes. With a few exceptions, IMPATT diodes have been restricted to Si, GaAs or InP, although IMPATT devices using different materials for the avalanche and drift regions have been reported. Higher power and lower noise were claimed for AlGaAs/GaAs heterojunction IMPATTs [J. C. Dejaeger, R. Kozlowski and G. Salmer, IEEE. Trans. Electron. Dev. 30, 790 (1983); Electron. Lett. 20, 803 (1984); M. J. Bailey, Microwave J. 36, 76 (1993); IEEE Trans. Electron. Dev. 39, 1829 (1992); M. J. Kearney, N. R. Couch, R. S. Smith and I. S. Stephens, J. Appl. Phys. 71, 4612 (1992)]. However other studies conclude there is little advantage for this material system [M. J. Kearney, N. R. Couch, J. Stephens and R. S. Smith, Semicond. Sci. Technol. 8, 560 (1993)]. Improved performance was also predicted for GaAs/InGaAs/GaAs IMPATTs [G. N. Dash and S. P. Pati, Appl. Phys. A, 58, 211 (1994)].
IMPATTs with superlattice avalanche regions have been proposed [C. C. Meng and H. R. Fetterman, Solid-State Electron. 36, 435 (1993); K. K. Chandramojhan, R. U. Khan and B. B. Pal, J. Inst. Electr. Telecom. Eng. 40, 261 (1994)], and some performance enhancement claimed [C. C. Meng, S. W. Siao, H. R. Fetterman, D. C. Streit, T. R. Block and Y. Saito, Micro. and Opt. Tech. Lett. 10, 4 (1995)].
Hence, in summary, materials for FET channels, HBT collectors, and IMPATT devices require high breakdown voltage and high saturated carrier velocity. APDs and IMPATTs require control of the ratio of the electron to hole ionisation coefficients (xcex1/xcex2). Thus, from the foregoing, it will be seen that there are a number of different conflicting design requirements for the impact ionisation and breakdown characteristics for semiconductor devices, which can only partially be resolved by conventional design techniques. These problems and constraints may at least in part be ameliorated by the present invention.
According to the present invention, there is provided a method of forming a semiconductor structure comprising forming a semiconductor structure such that the Brillouin-zone-averaged energy bandgap ( less than Ec greater than ) of at least a portion of the structure is controlled on the basis of a desired avalanche breakdown characteristic therefor. The present invention applies particularly to materials having relatively large bandgaps (Eg greater than 1 eV), including (but not limited to) GaAs, InP, GaP, AlAs, AlSb and AlP.
It has been found that for most practical purposes the following relationship holds:                                           1            8                    ⁢                      (                                          E                Γ                            +                              4.                ⁢                                  E                  L                                            +                              3.                ⁢                                  E                  X                                                      )                          =                  ⟨                      E            c                    ⟩                                    (        4        )            
where Excex93is the conduction band-edge at the xcex93 symmetry point of the Brillouin zone, measured relative to the valence band maximum at xcex93, EX is the conduction band-edge energy at the X symmetry point of the Brillouin zone, measured relative to the valence band maximum at xcex93, EL is the conduction band-edge energy at the L symmetry point of the Brillouin zone, measured relative to the valence band maximum at xcex93.
The portion of the structure may be formed according to the invention by:
(a) selecting from xe2x80x9cknownxe2x80x9d materials, such GaAs, InP, on the basis of their  less than Ec greater than  values which are fixed;
(b) forming an alloy with a predetermined value of  less than Ec greater than  and if required also a predetermined value of Eg or lattice constant;
(c) forming of a strained layer or strained layer superlattice such that a predetermined value for  less than Ec greater than  is established by the strain;
(d) forming a compositionally graded region in which the variation of  less than Ec greater than  with position (x) leads to an additional effective field d less than Ec greater than /dx, to enhance or suppress ionisation/breakdown depending on whether it is aligned parallel or anti-parallel to the applied electric field;
(e) forming a superlattice pseudoalloy; or
(f) forming a multilayer heterojunction system to create a superlattice or multiple quantum well, in which ionisation may be enhanced by means of the energy gained at the heterojunction in crossing from a region of high  less than Ec greater than  to a region of low  less than Ec greater than .
A device according to the invention may be formed using any combination of the methods (a) to (f) listed above.
Forming a compositionally graded region allows the possibility of engineering both  less than Ec greater than  and Eg such that either Eg is constant and  less than Ec greater than  is graded or  less than Ec greater than  is constant and Eg is graded.
It is known that strain in the lattice of a semiconductor also affects the band-structure. Thus, the avalanche breakdown characteristics can be tailored by growing mismatched lattice layer in the formation of a semiconductor structure. The control of EX and EL may also be achieved by varying the composition of the semiconductor material. Consequently, by the use of suitable lattice strains and material compositions,  less than Ec greater than  and Eg can both be tailored to the requirement of a particular device. Thus, it can be seen that the avalanche breakdown voltage or the ionisation coefficient for a junction can be varied somewhat independently of the energy bandgap Eg.
Grading  less than Ec greater than  or forming the structure with discontinuities in  less than Ec greater than  enables X and L electrons to be accelerated as well as the xcex93 electrons.
Conveniently, the Brillouin-zone-averaged energy bandgap of a p-i-n diode can be selected according to the following relationship:                               ⟨                      E            c                    ⟩                =                              (                                          V                b                            +              m                        )                    n                                    (        5        )            
where Vb is a predetermined breakdown voltage, and m and n constants. In the case of a p-i-n diode having a 1 xcexcm wide i region, m is approximately 45.8 and n is approximately 46.3. Other relationships between breakdown voltage Vb and  less than Ec greater than  can be calculated for other doping profiles, e.g. abrupt or graded.
The present invention also provides a semiconductor device having a region wherein the energy gap is substantially constant and the Brillouin-zone-averaged energy bandgap varies spatially and a semiconductor device having a region wherein the energy gap varies spatially and the Brillouin-zone-averaged energy bandgap is substantially constant.